Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates
Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang
TL;DR
The paper addresses quantum-state tomography for states prepared by Clifford circuits with $O(\log n)$ non-Clifford gates. It develops two efficient learning algorithms: one using entangled Bell measurements across two copies, and another using only single-copy measurements, both achieving $d_{tr}(\psi,\widehat\psi)\le \varepsilon$ with time poly$(n,2^t,1/\varepsilon)$ and copy complexity poly$(n,1/\varepsilon)$, under the assumption that the stabilizer-dimension is at least $n-t$. Central to the approach is reducing learning to identifying a large Pauli stabilizer group $G$, compressing non-Clifford content into $t$ qubits via a Clifford circuit $C$, and performing tomography on the small subsystem together with basis measurements to recover the full state. The work also extends to mixed states and to states with large stabilizer dimension, providing a robust property-testing tool for stabilizer-dimension and a constructive reduction that connects heavy-$p_\psi$/$q_\psi$-mass subspaces to efficient tomography. Together, these results yield near-optimal tomography for a broad class of states beyond stabilizer states, with potential practical impact for verifying and characterizing near-Clifford quantum devices.
Abstract
We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|ψ\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|ψ\rangle$ to learn $|ψ\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|ψ\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.